Theorem rnun 5029

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 5026	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4821	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4827	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2196	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4631	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4631	. . 3 |- ran A = dom `' A
7		df-rn 4631	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3285	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2206	1 |- ran ( A u. B ) = ( ran A u. ran B )

Syntax hints:   = wceq 1353   u. cun 3125   `'ccnv 4619   dom cdm 4620   ran crn 4621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-cnv 4628  df-dm 4630  df-rn 4631

This theorem is referenced by:  imaundi  5033  imaundir  5034  rnpropg  5100  fun  5380  foun  5472  fpr  5690  fprg  5691  sbthlemi6  6951  exmidfodomrlemim  7190